Abstract
Generalizing the work of Farahat-Higman on symmetric groups, we describe the structures of the even centers \(\mathcal{Z}_{n}\) of integral spin symmetric group superalgebras, which lead to universal algebras termed as the spin FH-algebras. A connection between the odd Jucys-Murphy elements and the Catalan numbers is developed and then used to determine the algebra generators of the spin FH-algebras and of the even centers \(\mathcal{Z}_{n}\) .
Article PDF
Similar content being viewed by others
References
Brundan, J., Kleshchev, A.: Representation theory of symmetric groups and their double covers. In: Groups, Combinatorics & Geometry, Durham, 2001, pp. 31–53. World Scientific, Singapore (2003)
Farahat, H., Higman, G.: The centres of symmetric group rings. Proc. Roy. Soc. (A) 250, 212–221 (1959)
Goulden, I., Jackson, D.: Combinatorial Enumeration. Series in Discrete Math. Wiley-Interscience, New York (1983)
Józefiak, T.: Characters of projective representations of symmetric groups. Expo. Math. 7, 193–247 (1989)
Józefiak, T.: Semisimple superalgebras. In: Algebra–Some Current Trends, Varna, 1986. Lect. Notes in Math., vol. 1352, pp. 96–113. Springer, Berlin (1988)
Jucys, A.: Symmetric polynomials and the center of the symmetric group rings. Rep. Math. Phys. 5, 107–112 (1974)
Kleshchev, A.: Linear and Projective Representations of Symmetric Groups. Cambridge Tracts in Mathematics, vol. 163. Cambridge University Press, Cambridge (2005)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Clarendon Press, Oxford (1995)
Murphy, G.: A new construction of Young’s seminormal representation of the symmetric group. J. Algebra 69, 287–291 (1981)
Murray, J.: Generators for the centre of the group algebra of a symmetric group. J. Algebra 271, 725–748 (2004)
Nazarov, M.: Young’s symmetrizers for projective representations of the symmetric group. Adv. Math. 127, 190–257 (1997)
Schur, I.: Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 139, 155–250 (1911)
Sergeev, A.: The Howe duality and the projective representations of symmetric groups. Represent. Theory 3, 416–434 (1999)
Wang, W.: The Farahat-Higman ring of wreath products and Hilbert schemes. Adv. Math. 187, 417–446 (2004)
Wang, W.: Universal rings arising in geometry and group theory. In: Cutkosky, S.D., Edidin, D., Qin, Z., Zhang, Q. (eds.) Vector Bundles and Representation Theory. Contemp. Math., vol. 322, pp. 125–140 (2003)
Wang, W.: Double affine Hecke algebras for the spin symmetric group. Preprint (2006). math.RT/0608074
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tysse, J., Wang, W. The centers of spin symmetric group algebras and Catalan numbers. J Algebr Comb 29, 175–193 (2009). https://doi.org/10.1007/s10801-008-0128-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-008-0128-1